Theoretical Calculations Are Converging On The Measured Muon g-2 Result

Background

Two high precision efforts have been made to measure a property of muons which are identical to electrons but more massive, called the anomalous magnetic moment of the muon called muon g-2, and the two experimental results largely confirm each other. A few years from now, a third precision measurement of muon g-2 will be made, but the expectation that it will confirm the two prior high precision measurements of muon g-2 is great.

When the new muon g-2 measurement results were announced on April 7, 2021, there were two leading calculations of what muon g-2 should be in the Standard Model of Particle Physics that differed in how the value they assigned to the indirect QCD (i.e. strong force) contribution to this observable measured to extreme precision.

One calculation from the Theory Initiative, supported by the group that conducted the latest measurement, shows a highly significant deviation from the experimentally measured value based upon a mix of different experimental measurements and actual QCD calculations.

The other calculation by the BMW group was consistent with the experimental measurement and used more or less pure lattice QCD calculations from first principles without substituting experimental results for parts of the QCD calculations. The BMW group asserts that the Theory Initiative didn't do this properly.

It also needs to be said that both prediction for the value of and both precision experimental measurements of the muon anomalous magnetic moment (without the -2 modification or the customary division by two) are identical when rounded to the first nine digits (basically, up to the parts per billion level). The range of values is: 

2.002 331 84 + 0.000 000 002 - 0.000 000 004. 

It is only the extreme precision of both the measurements and the ability to calculate a predicted value of this quantity in the Standard Model that makes it possible to notice than any of these four values are significantly different from any other of them. This open question in physics is being explored against a background of wide agreement on most points. The differences boil down to modest differences in how to conduct the QCD part of the calculation which makes only a small contribution to the total value of the muon anomalous magnetic moment.

So, even if the Standard Model is missing some new physics that is reflected in the observed value of muon g-2 the magnitude of the new physics must be quite small.

All of this background can be summed up in the following chart:

Where Are We Eighteen Months Later?

Increasingly, more than a year and a half later: "Our new lattice calculations are making it more apparent that the theoretical prediction [for the value of the muon g-2] is likely to move closer to the measured result." 

This statement is based largely on multiple papers looking at the HVP window (a subpart of the HVP calculation) this summer.

In other words, the BMW calculation is increasingly looking more correct than the Theory Initiative calculation.

What's At Stake?

The stakes are high because the measured value of muon g-2 is powerful global measure of the extent to which all components of the Standard Model combined work together to reproduce a highly precisely measurable observable quantity.

A significant experimental deviation from the correct Standard Model calculation of muon g-2 implies that beyond the Standard Model physics must exist and approximates the magnitude of those BSM physics effects.

But, if the correct Standard Model calculation of muon g-2 is consistent with the experimental measurement, then a huge swath of possible deviations from the Standard Model are ruled out. 

If the measured value of muon g-2 is consistent with a correctly calculated value of muon g-2 in the Standard Model, then only BSM effects that cancel out in the muon g-2 calculation, or BSM effects have a negligible impact on the muon g-2 calculation (essentially only extremely high energy effects) would be consistent with this experimental result. 

A muon g-2 measurement that is consistent with a correct Standard Model calculation of muon g-2 is particularly disappointing for anyone hoping to see BSM physics at a next generation particle collider, because it is much more heavily influenced by new physics that are just out of reach, than it is by new physics that only manifest significantly at energies far beyond the energy scales of even a next generation particle collider. 

Even if the muon g-2 is not perfectly consistent with the correct Standard Model calculation of muon g-2, the smaller the difference is the more subtle or remote in energy scale the new physics giving rise to the discrepancy must be. So, every time the theoretical calculation of the Standard Model prediction for muon g-2 gets closer to the experimentally measured result, the expected magnitude of any new physics gets smaller and harder to observe directly.

The current Large Hadron Collider experiment meanwhile, has so far failed to detect any really significant new physics or anomalies other than inconclusive hints that a property of the three charged leptons: electrons, muons, and tau leptons, called "lepton universality" may not be correct. 

But this weak force driven effect, by itself, even if it is real, would be smaller than what the Theory Initiative claims for the muon g-2 anomaly, because the weak force contribution to the value of muon g-2 is small and very precisely calculated with validating measurements in other contexts.

The absence of any observed BSM physics that would affect muon g-2 at the LHC combined with an experimental value of muon g-2 consistent with the correctly calculated value of muon g-2 in the Standard Model, would make the prospects of a new physics discovery at a next generation particle collider weak indeed, which undermined the science justification for building it right away.

The lack of new physics other than the Higgs boson at the LHC and a lack of indications of new particle physics from other sources like the muon g-2 measurement is what physicists sometimes call the "nightmare scenario" because it means that they have no new physics to discover and win Nobel prizes for themselves for discovering.

The Implied Magnitude Of The Hadronic Component

Muon g-2 the product of a pure QED component (which is predominant and ultra-precisely calcuated), an electroweak force component (which is the smallest contribution but precisely calculated), and a hadronic (strong force) component.

The combined result of the experimental measurements of muon g-2 (all of the numbers that follow are in the conventional -2 and divided by two form times 10^-11) is:

116,592,061.00(4100).

The QED + EW  predicted value is:

116,584,872.53(101) 

About 99% of the combined uncertainty in this value is from the EW component.

The difference, which is the experimentally implied hadronic component value, is:

7,188.47(4101)

This has a plus or minus two sigma range of:

7,106.45 to 7,270.49

The hadronic QCD component is the sum of two parts, the hadronic vacuum polarization (HVP) and the hadronic light by light (HLbL) components.

In the Theory Initiative analysis the QCD amount is 6937(44) which is broken out as HVP = 6845(40), which is a 0.6% relative error and HLbL = 98(18), which is a 20% relative error. But the stated uncertainty in the Theory Initiative prediction is almost surely greatly understated.

Fermilab (2021): 116,592,040(54)
Brookhaven's E821 (2006): 116,592,089(63)
Combined measurement: 116,592,061(41)
Theory Initiative calculation: 116,591,810(43)
BMW calculation: 116,591,954(55)

This implies that the BMW total hadronic value is about 7082(55), a difference from the Theory Initiative calculation that is entirely or predominantly due to differences in the HVP portion of the calculation ("here we use ab initio quantum chromodynamics (QCD) and quantum electrodynamics simulations to compute the LO-HVP contribution."). The open access version of the BMW paper here states that BMW leading order HVP value is 7075(55).  The methodology of their result and approaches to improving it was discussed here in light of the new experimental muon g-2 measurement.

I'm not certain how the incorporation of non-leading order HVP contributions and the HLbL portions of the QCD contribution give rise to the BMW combined muon g-2 cited above. The following chart is from the BMW paper:

Recall also that on the day that the new muon g-2 experimental results was released a "new calculation of the hadronic light by light contribution to the muon g-2 calculation was also released on arXiv . . . (and doesn't seem to be part of the BMW calculation). This increases the contribution from that component from 92(18) x 10^-11 . . . to 106.8(14.7) x 10^-11."

This boost of 14.8 in the overall QCD component isn't as big as the BMW HVP calculation's impact on it, but the two combined narrow the gap even more.